Consider the following form for the Navier-Stokes equations for a compressible fluid in steady-state:
$$div(\rho \textbf v)=0$$ $$div(\rho \textbf{vv} )=-k^2 \nabla \rho$$
where k is a constant. If the density is a known function of 3D space, is it possible to solve for the velocity function $\bf{v}$? The type of density functions I'm interested in would be of the form: $$\rho=\frac{c}{(R^2 + x^2 +y^2+z^2)^{n/2}}$$ where n is an integer.